Misconceptions of Universe Expansion, Accelerated Universe Expansion, and Their Sources. Virtual Reality of Inflationary Cosmology

In this work, we present our theory and principles of the mathematical foundations of Lobachevskian (hyperbolic) astrophysics and cosmology which follow from a mathematical interpretation of experimental data in a Lobachevskian non-expanding Universe. Several new scientific formulas of practical significance for astrophysics, astronomy, and cosmology are presented. A new method of calculating (from experimental data) the curvature of a Lobachevskian Universe is given, resulting in an estimated curvature-K on the order of 10 m. Our model also estimates the radius of the non-expanding Lobachevskian Universe in a Poincare ball model as approximately 14.9 bly. A rigorous theoretical explanation in terms of the fixed Lobachevskian geometry of a non-expanding Universe is provided for experimental data acquired in the Supernova Project, showing an excellent agreement between experimental data and our theoretical formulas. We present new geometric equations relating brightness dimming and redshift, and employ them to fully explain the erroneous reasoning and erroneous conclusions of Perlmutter, Schmidt, Riess and the 2011 Nobel Prize Committee regarding “accelerated expansion” of the Universe. We demonstrate that experimental data acquired in deep space astrophysics when interpreted in terms of Euclidean geometry will result in illusions of space expansion: an illusion of “linear space expansion”—Hubble, and an illusion of “accelerated (non-linear) space expansion”—Perlmutter, Schmidt, Riess.


Motivation and Background
The motivation for the present paper is to give a rational, in terms of clearly defined mathematics, explanations of some deep space astrophysical data. By rational, we mean that we only use fixed Lobachevskian (hyperbolic) geometry and not the pseudoscience of space expansion.
This paper can be regarded as an instance of physical Lobachevskian geometry.
For a reader new to Lobachevskian geometry, it would be beneficial to get some basic information on the subject from the following sources: [1]- [6].
The aim of the present work is twofold: 1) To give a rigorous mathematical analysis of data acquired in deep space astronomy and astrophysics.
2) To dismiss the claim of Perlmutter, Schmidt and Riess about the "accelerated expansion of space", and to stop the "expanding Universe".
We entirely reject the misconception of space expansion as having no experimental evidence. In a series of our work on applications of Lobachevskian geometry to physics, astrophysics, and cosmology [7]- [12], (see also Kadomtsev [13]), we proved that a non-expanding Lobachevskian Universe, being a Lorentz invariant entity, is able to explain in a coherent, Lorentz invariant way, all electromagnetic observable phenomena, including cosmological redshift, Doppler shifts, non-Euclidean intensity fading, space caused aberration, velocity caused aberration, polarization of light rotation, and CMB. Our exposition is done exclusively within the framework of Lobachevskian geometry and Maxwellian electromagnetics, which both are clearly defined and well understood.

Major Results of This Work
The major scientific results of this work are as follows: 1) A geometric brightness dimming parameter β (a first of its kind in science) is defined and introduced as a measure of light dimming in the Lobachevskian Universe.
2) An equation relating Lobachevski-von Brzeski cosmological redshift z and Lobachevski-von Brzeski intensity of light dimming parameter β is presented. Its linearized version gives a perfect fit to the experimental redshift/brightness data at low z acquired in the Supernova Project.
3) A method is proposed to calculate the curvature of the Lobachevskian Universe from the photometry data via the above parameter β. As an illustration, we estimate the Gaussian curvature K of Lobachevskian Universe to be

G. von Brzeski, V. von Brzeski
Agreement of geometry and physics is perfect.
6) The focusing properties of the Lobachevskian Universe are analyzed and favorably compared to astrophysical experimental data.
The content of this paper can be regarded as physical Lobachevskian geometry, which is actually a "translation" of statements and formulas of Lobachevskian geometry into the language of physics. This allows us to explain (in every detail) all phenomena related to light propagation in Lobachevskian space in a rigorous and logical manner, and in particular to explain why the Supernova Project conclusions are wrong.

Definitions
We work with real geometries and real 3 dimensional spaces only. By space, we mean a simply connected, Hausdorff, topological space.
of a set S (finite or not) and a family T of open sets covering S, called a topology on X.
The aim of topology is to formalize an intuitive notion of nearness between the points of a set S. A topological space, with a topology induced by a metrical structure, is called a metric space.
Having a distance structure on a topological space, we can interpret experimental data solely in terms of distances and appropriate functions of distances. This is a global look on geometry which we follow in our exposition. Definition 2 Lobachevskian (3 dimensional) real space is a simply connected, non-compact, locally compact, metric space of constant negative Gaussian curvature 0 K < . Definition 3 (Global). Euclidean space is the zero curvature, 0 K = , limit of Lobachevskian space.
On a local scales, Lobachevskian geometry can be approximated (with an arbitrary degree of precision determined only by the sensitivity our instruments) by Euclidean geometry. This is the linearization (i.e. Euclideanization) of Lobachevskian geometry. isometric. Physics in isomorphic spaces is qualitatively (but not quantitatively) the same, meaning it is isomorphic but not isometric. We also note that Maxwell's equations, which govern classical electromagnetic phenomena, are Lorentz invariant ( ( ) 2 PSL C invariant). We want to be understood not only by experts but also by junior students in physics, astrophysics, astronomers and by all those with an isnterest in Nature.
Thus, we use simple Lobachevskian geometry which can be easily comprehended.
We use standard notation. The word "light" in this paper means any electromagnetic radiation over the entire EM spectrum. So, for instance radio waves as well as gamma rays are understood as light. We limit our analysis to Lobachevskian physics of light. Other effects imposed on light, e.g. by gravitation, scattering, absorption etc., are not discussed here; however in real life, their impact on experimental data at telescope has to be evaluated.

Relation between Euclidean and Lobachevskian Geometry
Note that first of all, Euclidean geometry is the zero curvature limit of Lobachevskian geometry and all formulas of Euclidean geometry can be obtained in this limit. But some formulas of Lobachevskian geometry have no Euclidean counterpart. In this sense, Lobachevskian geometry is richer in content than Euclidean geometry.
Regarding the notion of distance, we note that in all formulas of Lobachevskian geometry, "distance" d never enters alone but only as a product of curvature K times distance d, or as the ratio of distance d to characteristic constant κ , which sets the absolute length scale in Lobachevskian geometry.
In Euclidean physics distances are in arbitrary units and therefore they are meaningless. The unit of the meter for instance, as well as all physical units in Euclidean geometry and Euclidean physics are brought from the "outside" of the system. Contrary to Euclidean geometry, in Lobachevskian geometry there is a natural unit of length defined in the following way (other choices are possible [10]).
Definition 4 The unit of length. We say that two parallel horospheres (horocycles in 2D) are at a unit distance 1 d = if the ratio of respective segments cut on them by two parallel geodesics is equal to e, the base of the natural logarithm.
Another distinctive feature of Lobachevskian geometry, contrary to Euclidean geometry, is the dependence of angles on distances. This is usually expressed From the above equations on volume and surface area growth in Lobachevskian space, it follows: Conclusion 5 About brightness data: The apparent brightness of a luminous object immersed in Lobachevskian Universe analyzed in terms of Euclidean astronomy (geometry) will appear to be dimmer than expected and consequently it will cause the illusion of being "more distant".
As we show in Section 4, Conclusion 5 states exactly what Perlmutter, Schmidt and Riess recorded. Conclusion 5 also states what causes the illusion of lower brightness and larger distance, misinterpreted as the effect of "space expanding" and pushing the object further away.
Sometimes one may encounter the question which geometry, Euclidean or Lobachevskian, is the "true" geometry representing the physical world? Or which geometry is "better"? The latter question is not appropriate since, as a mathematical systems, both geometries are equally valid. The first question depends on the particular situation in which the applicability of one geometry is justified more than of the other. This is determined by the sensitivity of our instruments which can or cannot to detect any deviation from Euclidean geometry in the problem of interest [5]. When modeling of the physical world, when geometric effects due to the negative curvature of Lobachevskian geometry are beyond the detection ability (sensitivity) of our instruments, we use Euclidean geometry instead of Lobachevskian geometry (which can be highly counterintuitive).
If by some "lucky accident" Lobachevskian (hyperbolic) geometry was developed ahead of Euclidean geometry, Euclidean geometry would follow instantly from it as it's zero curvature limit. In the opposite direction, from Euclidean to Lobachevskian geometry, it took 2000 years to recover the information lost in that zero curvature limit. This asymmetry is quite impressive.

Mathematics of Cosmological Redshift in a Non-Expanding Lobachevskian Universe. Its Geometric Origin, Properties, and Applications to Real Astrophysical Data
Lobachevskian geometry is inherently intertwined with electromagnetism. It affects all of the characteristics of electromagnetic radiation which propagates in Lobachevskian space. Light (i.e. any EM radiation) propagating in Lobachevskian space will experience space caused aberration (an effect analogous to velocity caused aberration), and a change in its frequency, intensity, and polarization. It is therefore important to understand how Lobachevskian geometry affects the characteristics of electromagnetic radiation, something of fundamental importance to astronomical and astrophysical observations. In the following sections, we will discuss the two effects of utmost significance to astrophysics data: cosmological redshift and apparent brightness. It is easy to see how in a few steps, physics can be extracted from geometry.

Light Frequency Shift as a Function of Distance and Curvature In Lobachevskian Geometry
Equation (1) tells us that two light rays separated by a distance 1 l at the source will be separated at a distance 2 l at the detector, 2 1 l l > . Distances 1 l and 2 l are measured along the horospheres orthogonal to light rays. Separation of horospheres is equal to s (which is the distance between the source and the detector measured as the geodesic parameter along the geodesic). Since on the horospheres the geometry is Euclidean, distances 1 l and 2 l are measured according to Euclidean geometry, while s is Lobachevskian (hyperbolic) distance. The constant κ sets the length scale in Lobachevskian space.
Given cosmological distances, we cannot "go" to the source, e.g. a Supernova 7-bly away, and to measure the separation between the two selected parallel geodesics. Instead we use light itself as the gauge plus Equation (1) (assuming that all atoms of the same sort in the Universe, under the same conditions, radiate the same spectrum). Therefore we select two geodesics separated at a source by a distance equal to some wavelength (green for instance) 1 At the detector, accordingly to fundamental formula of Lobachevskian geometry (1), we will detect the separation of geodesics, 2 2 1 l λ λ = > . Therefore: Equation (2) We call the geometric redshift (4) as Lobachevski-von Brzeski redshift G LvB z z = since the basis for it is Lobachevskian geometry and it was discovered and reported by G. von Brzeski, and von Brzeski V., in the works we listed above.
One may easily check that a linearized form of (4) yields: is the Gaussian curvature of Lobachevskian Universe.
To simplify the exposition, in all what follows, we use the following notation. We define K − as the negative Gaussian curvature of Lobachevskian space. Since 0 K − > , 0 K K − = > , and we will use K (positive value) as such everywhere below. The linear approximation (5) of our redshift Formula (4) (which holds for all z) recovers the original Hubble experimental data where distance is proportional to redshift. Note that the velocity space in Einstein's SR is also a Lobachevskian 3-dimensional real space metrized by relative velocities. The signed distance in Lobachevskian velocity space is simply relative velocity. Its Poincare ball model has a constant Gaussian curvature Therefore the same effect of frequency redshift (relative velocity outward) trivially follows from (4) as: which is the common Doppler Shift (for the Doppler blue-shift, i.e. relative velocity inward). We call kinematic frequency shifts as Lobachevski-Doppler shifts K lD z z = . Its linearized version is: The linear approximation (7) of (6) is routinely and intentionally misinterpreted in all literature via the Doppler effect as "proof" that Hubble observed receding galaxies [14]; see Appendix.

Taylor Series Approximation of Fundamental Equation of Lobachevskian Geometry Demonstrates the Illusion of Space Expansion
Since nearly all of the information we gather about the Universe comes to us via electromagnetic radiation-the geometry of light rays, the geometry of geodesics directly affects our observations. This is extremely important to realize since as our instrumentation methods get more and more powerful, we are able to collect information about behavior of geodesics on cosmological scales, information that was not accessible to astronomers of the past. The geodesic light ray in a Lobachevskian Universe runs over arbitrarily large distances. To analyze its behavior in terms of experimental science we need to look at it in a "step by step" fashion in accordance with the development of astronomical observations which only in the last 100 -200 years started to reach deep space ranges. We want to establish some qualitative relation between distance (geodesic parameter) from a luminous object to us and the power of our telescopes. Therefore we expand the geodesic light ray into a power series, and then each term of the expansion will correspond to the increasing power of our instruments to collect data from increasingly distant objects. This is shown in Figures 1-6.
(apparent brightness), with error beyond the resolution of our instruments, has to be regarded as "local" or Euclidean, and justifies the use of Euclidean geometry in such a region.  . The above figure shows the misrepresentation of astrophysical and astronomical data due to Hubble and all thereafter for the past 100 years. This is the extragalactic distance case, region B. The geometric spectral shifts are now within the detection power of diffraction gratings of telescopes. Parallel rays are no longer parallel in the sense of Euclidean geometry. They are parallel in the sense of Lobachevskian geometry. Viewed in terms of Euclidean geometry, they create illusion of "linear space expansion" in the above approximation.  Figure 3. This is the weakly Lobachevskian case interpreted erroneously as "linear space expansion". The separation of geodesics in this case is a linear function of distance along the ray. The step function of Euclidean parallel segments x ∆ (pieces of Euclidean geodesics) has constant vertical step y ∆ . It creates an illusion of "linear space expansion". Space does not expand; Lobachevskian geometry is fixed, and parallel geodesics spreads apart due to (1).  The separation of geodesics in this approximation is a quadratic function of distance, which is an elementary definition of uniformly accelerated motion. The above illusion was interpreted by Perlmutter, Schmidt, and Riess as "accelerated space expansion" and endorsed by the Swedish Royal Academy Nobel Prize Committee in 2011 [16].
Again we start from the fundamental equation of Lobachevskian geometry: The characteristic constant κ is related to the Gaussian curvature K as 2 κ − .
Since the characteristic constant κ (and consequently the curvature K is so far unknown, we set 1 K κ = = , which is a generic (arbitrary) value and has no effect on what follows. We estimate the value of K in section 5. We also set The following is important in order to understand the essence of the erroneous interpretation done by Perlmutter, Schmidt, and Riess, as well as by the Nobel Prize Committee awarding its 2011 prize for for that erroneous conclusion.
We analyze (9) as follows. First we write the expansion of a geodesic in a Lobachevskian Universe (9) in a power series (10). Next, by differentiating the power series term by term we obviously end up with the same exponential function (11). But now the the RHS of (11) is equal to the rate of change of distance ( ) s δ with a real parameter s. We see that at which the parallel geodesics (parallel light rays) are spreading apart.
Note that both expansions (10) and (11) start at a star location from which geodesic parameter is counted.
is the rate of change of the redshift with geodesic parameter (distance along geodesic). Equations (10) and (11) explain the most important events in the entire history of astronomy.
In accordance with (11)  B s is the numerical value of the local slope of a geodesic light ray in Lobachevskian space / Lobachevskian Universe. It is a non-negative real number which classifies the physical geometry applicable to astrophysical measurements and data. Remark 7 Our definition of redshift above has far reaching significance beyond cosmology. It applies to an entire new area of physics. New age meta-materials with sufficiently large engineered ( ) B s will enable X-ray and γ-ray frequency down conversion. This will allow us to see images from the interior of atomic nuclei and images of the interiors of stars and galactic nuclei where γ and X-rays are generated.

From Ancient Times to Pre-Hubble. First Term in the Power Series: Euclidean Astronomy With No "Expansion of Space"
This period is characterized by observations conducted by the naked eye and weak telescopes, limited to what we call region A in space. In a region A, the distance between the object and observer is "insignificant", meaning its linear size can be as large as 10 5 and perhaps 10 6 light years [5]. Thus we approximate e s by the first term in (10), and the RHS of (10) immediately tells us that redshift in region A (due to geometry) is trivially equal to zero.
( ) e 1 The speed of geodesic separation, which is given by the von Brzeski redshift parameter ( ) B s corresponding to the first term in (10) and shown by the first term in (11), is equal to zero.
Equation (13) states: a light ray in Lobachevskian space, in an approximation via the first term of the power series (10), is a constant function. This is a straight Euclidean line 1 γ = , i.e. a Euclidean geodesic. Conclusion 8 In approximation (12), light rays (exponentials) in Lobachevskian Universe are replaced by Euclidean straight lines.
Equation (13)  Conclusion 9 A separation speed of zero, ( ) 0 B s = , implies that geodesics are not spreading apart, which means that the distance between them is constant, or that they are parallel in the sense of Euclidean geometry.
Conclusions (8) and (9) state that the geometry of space recovered from expansions (10,11) in approximation (12) is Euclidean, which is in full agreement with the statement that locally (i.e. region A), Lobachevskian geometry may be approximated by Euclidean geometry.
In fact, using solely a geometric argument, we proved the following theorem: Theorem 10 von Brzeski G., von Brzeski V. Euclidean geometry of space cannot change the frequency of light propagating in it. Cosmological redshift due to Euclidean geometry of space is impossible.  Classical view: An electromagnetic wave propagating in Euclidean space has the same frequency 0 k and polarization (since 0 k = ±k ) at any point in its path.
 Quantum view: A photon propagating in Euclidean space has the same energy 0 k  and momentum k at any point in its path. Conclusion 11 If in a region of space regarded as Euclidean, spectral shifts are recorded, then their origin is due to relative velocity (Doppler), gravity, scattering and perhaps to other unaccounted factors.
The size of a region A is determined (at a fixed curvature) only by the sensitivity of our instrumentation. Today, with modern diffraction gratings with a resolving power of 10 6 , region A means we operate in a domain of 10 5 light-years(size of the galaxy) and perhaps as much as 10 7 light-years in linear size. As long as astronomical observations were conducted within our galaxy, no systematic spectral shifts have been recorded and this is why do not see "space expansion" within our galaxy. Given that apparatus which records spectral shifts (diffraction grating) has a limited resolving power, and the spectral shift is a function of the distance d times Gaussian curvature K , cosmological distances are needed (at constant curvature) to make it possible for diffraction gratings to "see" it. Make a diffraction grating with a resolving power, say 10 15 and you will see cosmological redshift in our galaxy, contrary to what cosmologists tell us. Make a diffraction grating with a resolving power of 10 20 and you will record cosmological redshift in your own backyard. The above analysis shows (see Figure 1 and Figure 2) that on galactic distance scales in a Lobachevskian Universe, the von Brzeski redshift parameter ( ) 0 B s = , which implies Euclidean geometry. This is the reason that we do not see any illusion of "expanding space" on galactic scales.

Post-Hubble Epoch. Better Telescopes See Now See the Second Term in the Power Series
This is the period of the illusion of "linear space expansion". It spans the time period from Hubble's mistake on extra galactic redshift [14] to the erroneous Supernova project conclusions by Perlmutter, Schmidt, Riess [15]. In this period, improved telescopes are able to acquire light from more distant objects in space. This corresponds to approximating the light ray e s by the first expansion term in (10) which is s: We see that in approximation (14) redshift z is proportional to the geodesic parameter (the distance along geodesic). This is exactly what Hubble observed, but not what is advocated in all literature as the "Hubble distance velocity law". Does that mean that space is undergoing expansion and/or inflation? Of course not. However, Hubble did not know Lobachevskian geometry and he misinterpreted the linear approximation of the piece of the exponential geodesic in Lobachevskian universe as a straight Euclidean geodesic, plus "linear inflation". The reader should be aware that the proportionality of measured redshift versus distance (as measured by Hubble), which results from approximation (14), is notoriously misrepresented in all literature as "'proof" that Hubble experimentally observed receding galaxies and space inflation; see e.g. Perlmutter [15] for that misrepresentation. We return again to Edwin Hubble's mistake in Section 8. For speed of geodesic separation in approximation (14) we have: which reads that speed of geodesic separation is constant, and the redshift z is

From the Mistake of Perlmutter, Schmidt, Riess to Present. More Powerful Optics Now See the Third Term in the Power Series
This is the period of the illusion of "non-linear, accelerated space expansion", in which we discuss where the conclusion of "accelerated expansion" came from.
Progress in technology resulted in advanced telescopes able to see the third term of a power series (10), i.e. able to acquire data from objects located in deep space, say From approximation (16), we see that separation δ is speeding up in a non-linear fashion, and the redshift now is also a non-linear function of distance.
Again, the speed of separation ( ) B s is given by (11) and it is shown on We know that if the speed (rate of change) is proportional to the geodesic parameter, such motion is called uniformly accelerated motion. In region C, the slope ( ) B s appears closer to an exponential function, and the distance between the reference light ray and the ray parallel to it blows up very rapidly. The Euclidean distance between the steps increases in non-linear (seemingly "accelerated") fashion.
There is no principal difference at all in tracing the geodesic light ray ( ) s γ in a Lobachevskian Universe or in drawing (tracing) the geodesic ( ) s γ on a piece two dimensional space (a sheet of paper)-the flat-by a pencil. (The flat, or totally geodesic surface, is an analogue of the Euclidean plane. It has the property that if a geodesic has two common points with the flat then the geodesic lies entirely in the flat.) We hope that nobody will claim that geodesics are spreading because the piece of paper is expanding. However, the "official" version of cosmological redshift which is currently in all sources on cosmology, presented here by Nobel Prize Laureate Saul Perlmutter, states: "...the redshift, is a very direct measurement of the relative expansion of the universe, because as the universe expands the wavelengths of the photons traveling to us stretch exactly proportionately-and that is the redshift" [15].
We strongly disagree with the above definition of frequency shifts recorded at the telescope. As we explained already (see Equations (4) and (6)), the recorded shift in frequency (ignoring other factors) is a mixture of unknown proportions of a systematic geometric redshift (4) and kinematic, random blue / red Doppler shift. This is shown in detail from classical, quantum, and topological points of view in the Appendix.
Putting the fundamental error of space expansion at the base of all of their data reduction process, Perlmutter, Schmidt and Riess automatically nullified the entire conclusion of their subsequent work. This will be clearly seen below, where we calculate the relation between photometry and redshift in a Lobachevskian Universe.

Photometric Data Analysis in Supernova Project and the Sources of Its Failed Interpretation
In Note that the surface of a Lobachevskian sphere grows like e r so the apparent brightness will be substantially less than the apparent brightness expected in Euclidean astronomy. This factor alone explains why Perlmutter, Schmidt, and Riess recorded 25% lower brightness than expected in the Euclidean astronomy they used [15] (no space inflation needed).
2) Non-Euclidean-Lobachevskian geometric dimming (decrease in intensity) -see von Brzeski G., von Brzeski, V. [8]-is analogous to a geometric decrease in frequency or geometric redshift. Geometric dimming of brightness is a monotonically decreasing function of Lobachevskian distance asymptotically approaching zero as the distance increases to infinity. Points at an infinite distance from any point in the Lobachevskian Universe belong to the boundary at infinity of the Lobachevskian Universe. We will never see any object at the boundary at infinity in Lobachevskian Universe due to this effect alone. It manifests itself as a dark background-the dark night sky.
3) The apparent brightness I and apparent color (frequency) are both affected by geometric and kinematic factors (as discussed in the Appendix and brighter, while a star moving away will appear reddish and dimmer [8] provided that the distances are the "same". The behavior of light in a Lobachevskian Universe is affected by the negative curvature of a large scale vacuum and by the the negative curvature of Lobachevskian velocity space.
Scattering, dust, etc., are not of interest to us. Now we show in detail the nature of the fallacious reasoning and errors of Perlmutter, Schmidt, Riess, and the Swedish Royal Academy, which led to "accelerated space expansion".
We recall that the surface of a sphere in a Lobachevskian Universe is given as where r is the Lobachevskian radius of the sphere and κ is the characteristic length constant setting the absolute length scale related to the (negative) Gaussian curvature as 2 1 K κ = .
We expand (19) in a power series around zero (or origin/center of the sphere) where the light source (e.g. supernova) is located. This is because we want to see what brightness will look like with respect to each term in the expansion. Note that in the realm of the Lobachevskian Universe, "around zero" might be a few hundred light years or so. Since  Perlmutter, Schmidt, and Riess had properly interpreted Lobachevskian data, then all the "surprises" would disappear and agreement between experimental data and and Lobachevskian geometry of the Universe would be perfect.
However that was not the case, and in his paper [15], Perlmutter argues that "while light traveled billions of years to reach the telescope, the (Euclidean) Universe was constantly expanding, so the sphere 2 4π S R = expanded as well". As a result, at the time when the light reached the telescope, the density of light for a larger, expanded Euclidean sphere would obviously be lower.
However, as it turned out, the present value of "universe expansion" was enough to fit the data. Therefore, Perlmutter arbitrarily accelerated the entire infinite universe to obtain a fit to the 2 1 4πr law of Euclidean photometry.
Also let us recall the announcement of the Royal Swedish Academy [16] regarding 2011 Nobel Prize in Physics: "By comparing the brightness of distant, far away supernovae with the brightness of nearby supernovae, the scientists discovered that the far away supernovae were about 25% too faint. They were too far away. The Universe was accelerating. And so the discovery is fundamental and a milestone for cosmology. And a challenge for generations of scientists to come".
It amazing that wording of the Royal Swedish Academy and the Nobel Prize Committee echoes our Conclusion 5 of this work, with one essential difference however. We proved that an observer, in a Lobachevskian universe, will record an anomalous (in terms of Euclidean photometry) light dimming and being ignorant of hyperbolic photometry laws, he/she will wrongly interpret the result as an effect of arbitrarily increased distance. The Nobel Prize Committee, due to their ignorance of Lobachevskian geometry, takes this illusion as a real thing.

Calculation of the Curvature of the Lobachevskian Universe
So far our geometric analysis was done in general terms which allowed us to make correct qualitative conclusions but no numerical results have been The following successful calculation of the negative curvature of the Universe from experimental data, resulting in credible numbers, is first such calculation in science.
Looking at expansion (20), we have learned that indeed the Lobachevskian sphere is larger than a Euclidean sphere of the same numerical radius, as we see in Figure 7. Now we rewrite the expansion in geometric form, which tells us about the physics of each term. We write expansion (20) as power series in terms of curvature, π π higher order terms 3 45 We see that the coefficient of K to the 0th power is simply the Euclidean sphere, which again tells us that Lobachevskian geometry appears to be flat locally, (no detectable curvature on local scales). As we move further away from the source of light, the curvature begins to be noticeable, and consecutive coefficients of powers of K show how much the area of the sphere grows relative to the Euclidean sphere. We also see that each term has dimension of length squared m 2 .
Given Equation (19), we now define a quantity entirely analogous to the Lobachevski-von Brzeski geometric redshift, which may be considered as the relative excess of length of the received wavelength (far from source), with respect to the wavelength close to the source. This change in wavelength, as we stated in (4), is a function of distance and curvature. This time however, we define the relative excess in area as ratio of the surface areas of a sphere far from the source (Lobachevskian sphere) relative to a sphere close to the source (Euclidean sphere). Obviously this ratio, call it DISTANT CLOSE , by photometry rules, is an inverse of the brightness ratio. We see that Definition 12 below (a) (b) Figure 7. Here we see two spheres of the same numerical radius r. The one denoted by A is the Euclidean sphere. Its surface grows as 2 r . The one denoted by B is the Lobachevskian sphere 2 L S . Its surface grows as e r . Space is neither expanding nor expanding in accelerated fashion; however the geometry of space which on local scales (small distances) appears to be Euclidean, is Lobachevskian on a global scale; see power series (10). The total luminous energy output of a distant source spread over the Lobachevskian sphere (whose volume is larger than its Euclidean counterpart) will result in "lower apparent brightness" than expected from Euclidean geometry. exactly fits with the data acquisition scheme in Supernova Project; see [15]. Definition 12. von Brzeski G., von Brzeski V.. The Lobachevski-von Brzeski brightness dimming parameter β is the inverse ratio of the DISTANT source apparent brightness to the CLOSE source apparent brightness, provided that luminosities of both are the same. In other words, β is the ratio of surfaces of the Lobachevskian sphere to the Euclidean sphere. The reader should note that the above very crude calculations nevertheless produce very reasonable numbers which cannot be ignored. The mathematical framework is correct but there is uncertainty in data due to several reasons. For instance we do not know the contribution to the measured brightness from kinematics (see the Appendix). True data will have to be harvested from a large number of equidistant sources (like a shell) which will average out kinematic brightness due to the random orientation of relative velocities.
We are also confident that calculations based on our geometric formula for cosmological redshift (4) will result in similar numbers, but redshift data needs to be genuine dimensionless data from instrumentation, not data that has been manipulated in terms of units of velocity. Nevertheless we invite all astrophysicists, astronomers, and all those with access to relevant data to work on mathematical applications of Lobachevskian geometry.
A precise determination of the curvature of the Lobachevskian Universe is a hard task. But if this were already done, then diffraction gratings and photometers could be promptly calibrated in units of distance and distances to all luminous objects in the universe will be known instantly, provided that all other factors (kinematic, scattering, etc.) are accounted for.

Lobachevski-von Brzeski Cosmology Versus Big Bang Accelerated Space Expansion
Perlmutter, Schmidt, and Riess's reasoning and conclusions about an accelerating expansion of the Universe can easily by traced to the 100-year old Hubble error linking redshift and the apparent recession velocity of galaxies, expressed via the fictitious relation called the "Hubble distance velocity law". See discussion of this in Section 8.
Being embedded in a Lobachevskian Universe, the high redshift supernovae obviously showed lower than expected (from Euclidean geometry) brightnesssee (19) and (21), therefore exhibiting an illusion of being further in Euclidean space. From the brightness-redshift relation (again in the Euclidean geometry of Big Bang cosmology), it follows that the dimmer the object, the higher the redshift. In the virtual world terminology of an "expanding Universe" redshift is given by another quantity called the Hubble constant H, which in reality has nothing with the fiction of space expansion; see section 8.
We now demonstrate how the redshift-brightness dimming parameter relation works. We write the Lobachevski-von Brzeski redshift Equation (3) Eliminating K from (28) and (29) we get a very important Lobachevski-von Brzeski Equation (30) between β and z (the first of its kind in science), provided that kinematic (and other) factors are not present or are known and accounted for.
Equation (30), discovered by us, for relative intensity down shift β versus spectral down shift (redshift) z, expressed in terms of pure real numbers, is an exact result. It represents a true law of Nature which reflects geometric relations between space and an electromagnetic field in a Lobachevskian Universe. This is how Nature works, and may serve as an illustration to what Eugene Wigner calls "an unreasonable effectiveness of mathematics in natural sciences".
We can easily get a linearized version (small redshifts-weakly Lobachevskian case) of Equation (30). Since: and since for small z: ( ) we have a linearized version of our general Equation (30) as: Note that in (33), β is proportional to z squared, which is consistent with the fact that β is the relative change in (surface) area whereas z is the relative change in wavelength (i.e. length).
It is interesting to note that the same result can be obtained in another way.
We note that the inverse of (29) is in fact the inverse of the expansion of sinh x x in the power series with coefficients called Bernoulli numbers n B : In (34), n B are Bernoulli numbers. For instance the first three are: 3 1 42 B = , and so on. The numbers n B were introduced to mathematics by Jacob Bernoulli Sr. in the 18th century. In the mid-20th century, they found an unexpected application in topology, in the theory of characteristic classes. Our case shows an unexpected applicability of Bernoulli numbers in Lobachevskian geometry and Lobachevskian astrophysics.
As an exercise, the reader may obtain the same result as (33), by noting that for small β, and taking into account only the first two terms of (34) i.e. the linearized, or weakly Lobachevskian case.
The astrophysicist familiar with Lobachevskian geometry will see in this the signature of the negative curvature of the Lobachevskian Universe. The astrophysicist ignorant of Lobachevskian geometry will claim that the rate of expansion was lower in the past than the present, past present H H < , since he associates expansion with redshift. Consequently, he will also claim that that expansion in the present is higher than in the past, Universe according to our Equation (29). As the plot clearly shows, there is a non-linear relationship between the dimming parameter β and redshift z: for a given increase in redshift, there is an even greater increase in dimming. This non-linear relationship was proclaimed by Perlmutter, Schmidt, Riess [15] and 2011 Nobel Prize Committee as the signature of "accelerated space expansion".

Hubble Constant as Measure of Gaussian Curvature of Non Expanding Lobachevskian Universe
Lets take a closer look at the so-called "Hubble constant" H. We now show that instead of being a measure of "space expansion" (which we believe does not exist), it is a measure of the Gaussian curvature of the Lobachevskian Universe. The "Hubble Constant" did not appear out of nothing. It came from real data acquired by Hubble [14] in a Lobachevskian Universe. However, due to Hubble's conceptual error (see Section 8), this data were misinterpreted in terms of velocity 1 z c υ = (7) instead of in terms of space 1 z d R = (5). We will show the true meaning of the Hubble constant in terms of a non-expanding Lobachevskian Universe. For instance, we take its "present value" as 75 k•ms −1 /Mpc. Since a megaparsec is such a huge distance that we cannot intuitively comprehend, we convert everything to meters (since 1 c = ). Taking  The fit between our theoretical calculations (25) based on a non-expanding Lobachevskian Universe cosmology and existing experimental data is excellent.
We proved that our model of a non-expanding Lobachevskian Universe works and delivers correct numbers. This is the way astrophysics and cosmology should follow. The Hubble constant belongs to Lobachevskian geometry and we indeed G. von Brzeski, V. von Brzeski live in Lobachevskian world.

A Global Look at the Lobachevskian Universe. A Concave Lens Model. Virtual Reality of Inflationary Cosmology
So far we have shown that Lobachevskian geometry applied to space yields logical and mathematically consistent representations of all phenomena concerned with the propagation of light on all range of distance. Nevertheless we have still not shown the virtual nature of the Big Bang. If a Lobachevskian geometry of space were unable to show that the Big Bang in reality never existed, this would be a substantial disadvantage. Fortunately the power of Lobachevskian geometry easily shows virtual nature (i.e. mirage) of the Big Bang, and all of the virtual science associated with it.
It is well known that a geodesic (light ray) passing in the vicinity of a massive body will be deflected inward due to the local positive curvature of space. This effect in geometrical optics is the same as deflecting geodesics (light rays) by a positively curved convex lens. In the same fashion, a geodesic will be continously deflected outward due to the global negative curvature in a Lobachevskian Universe. In geometrical optics this effect is the same as deflecting geodesics (light rays) by a negatively curved-concave lens. Lobachevskian lensing is a global effect, while gravitational lensing is a local effect. Thus one can think of gravitation as a local convex lens, and about Lobachevskian Universe as a globally concave lens. An observer who interprets his observations in terms of Euclidean geometry will be convinced that the object located at a real position a is at virtual (non-existing) position A. This illusion or mirage of "seeing" an object at a virtual position A is caused by an ignorance of the existing geometry of space between the object and an observer. Figure 9 shows this in detail: between the distant Universe and us the observers, a Lobachevskian space of constant negative curvature −K is "inserted", which spreads apart parallel geodesics (light rays), and all optical phenomena act in the exact same way as in a concave lens with the negative curvature.
We recall what Saul Permutter said about "looking backward in time with Hubble constant present value" [15]. Looking backward in time, Perlmutter reversed the sign of the geodesic parameter and ran the light rays back with the "present Hubble constant". Obviously, the backward-running virtual rays (showed as dashed lines in Figure 9) all converge in one point, which is the virtual focus of concave lens.
The distance from the virtual point of the Big Bang to an observer is called the Hubble radius of the Universe. The virtual time resulting from the division of virtual distance by the Hubble constant is called the "age of the Universe", another virtual entity. So when Perlmutter Schmidt and Riess were "looking backward in time", all they were seeing was the mirage of a virtual reality and the mirage of a virtually expanding Universe, a mirage produced by Lobachevskian geometry. The above analysis also shows the non-applicability of the notion of time in regard to the entire Universe.

The Sources of Difficulties in Nailing Down the Hubble Constant
The misinterpretation of the Hubble constant due to its identification with    This is the reason that some authors in the literature are trying to replace the Hubble constant with a "Hubble parameter".

"Dark Energy" or Negative Curvature of Lobachevskian Universe?
The concept of "dark energy" is a copycat version of Einstein's General Relativity (GR) for the negative curvature case. Its essence is as follows.
If the source of positively-bent space, or positive curvature, accordingly to GR is mass-energy density, then by analogy with real baryonic matter, there must be some, mysterious "dark matter energy density" pushing matter away and acting like some kind of anti-gravity, thus imposing a negative curvature 0 K < on space, and causing the mirage of space expansion. Lobachevskian geometry does not require any "dark energy", and we doubt there is any detectable "dark energy" in the Universe. The physical large scale vacuum is, simply speaking, negatively-bent. That's all. However those who love "dark energy" may identify it with the negative curvature of the Lobachevskian Universe.

Our Predictions Of Phenomena to Be Discovered in Astrophysics
Based on our previous papers and analysis of the interdependence of Lobachevskian geometry with electromagnetism, we predict the discovery of the following phenomena. These are already mathematically described in our papers listed above.
1) Lobachevskian space caused astronomical parallax. This parallax is different than the parallax in Euclidean geometry, since in Lobachevskian space an angle depends on distance and in Lobachevskian space the sum of the angles in a triangle is less than π.
3) Lobachevskian space caused change in polarization of light (magnitude and direction) or polarization rotation [8].
4) Lobachevskian space caused dimming of light, an effect analogous to cosmological geometric redshift, but affecting the intensity of light [8].
5) Detection of gravitational waves (GW), if any, due to the scattering of an electromagnetic wave on the variable local curvature of space, inflicted by GW and manifested by a periodicity of redshift and intensity of light described in [8]. The search and detection of GW proposed by us is based on the search of both amplitude and frequency modulated electromagnetic wave coming from some regions of interest in deep space. 6) Down conversion to the visible spectrum of X-ray and γ-rays based on geometric downshift of frequency of an electromagnetic wave (photons) in Lobachevskian meta-materials.
As we explained above, the above effects will be noticeable only at appropriately large distances when the negative curvature of Lobachevskian Universe will come into play. We invite astrophysicists to look for these effects since they significantly will improve our understanding of the Universe.

Summary
We note that Hubble, Perlmutter, Schmidt and Riess, as well as all other astronomers over the ages, have all been observing precisely the same reality: the Lobachevskian Universe, however at different scales determined by power of their telescopes. This progression is shown in Figures 1-6. 1) Hubble in 1922 saw the second, linear term s in the power series expansion (10) of the geodesic light ray e s in the Lobachevskian Universe. Being ignorant of Lobachevskian geometry, he could not understand (as all astronomers and astrophysicists of 20th century) that as the first man on the Earth, he directly experimentally experienced the Lobachevskian geometry of the Universe.
2) At the end of 20th century, Perlmutter, Schmidt and Riess, having better instruments, were able to look a bit further than Hubble. It follows that their telescopes were able to see the third, quadratic term The FRW metric and Einstein's equations are two logically and mathematically independent concepts. It is not granted a priori that their fusion will produce a meaningful result. The combination of the FRW metric (with a time dependent term) and Einstein's equations has led to an un-physical solution containing the mirage of "space expansion", characterized by the entity resulting from the error-the "Hubble constant" and associated wording.
Contrary to quantum mechanics, Einstein's GR did not result in some "breakthrough" in physics, neither on a micro scale nor on a global scale. On the micro scale, after a 100 years of intense effort, we still do not have the quantum theory of gravity and it is not clear is this possible at all. On the global scales,the questionable applicability of Einstein's general relativity to cosmology has not met expectations either. It is entirely unclear why a differential equation which in mathematics is considered to be a local object is suitable to describe the Universe in its infinite wholeness? Einstein's General Theory of Relativity (GR), while applicable to a star or galaxy, is entirely unsuitable as a tool for analyzing the Universe as a whole. It also should be said that any partial differential Equation (PDE) is mostly void unless initial and boundary conditions are not specified upfront. Initial and boundary conditions for the entire Universe (if any) are known perhaps to only God, if one exists. It is not surprising that in conjunction with the FRW metric, GR produced bizarre non-physical effects and conclusions of "space expansion", Big Bang, "space inflation", "dark energy", "accelerated space expansion".
We demonstrated in several ways that Lobachevskian space when interpreted in terms of Euclidean space will cause the illusion of "expansion of the Euclidean space". We represent it again, from ancient times up to present, in Figure 13, and reader can see himself/herself "expansion of space".
The illusions in the perception of space and motion (e.g. a flat Earth and Ptolemy's geocentric system) are not something unusual in the domain of science. The development of rational science may be regarded as a struggle to overcome false ideas and illusions, one after the other.
If Edwin Hubble and his fellow astrophysicists at the time knew the above geometrical formula of cosmological redshift (4) in its linear approximation (5), he obviously would have seen that it is exactly what he experimentally measured, Figure 13. The above figure again shows the illusions that result when Lobachevskian geometry is interpreted in terms of Euclidean (flat) geometry. Starting as parallel at the star, the distance the geodesics first spread apart in linear fashion, and then in non-linear fashion. This is the whole essence of the misunderstanding of space "expansion" and accelerated space "expansion". The newest "trick" in the idea of "space expansion" is the so-called "metric expansion", when the flat Euclidean Universe (space) expands into itself. The Lobachevskian geodesics (solid thin lines) are replaced by piecewise parallel (in a Euclidean sense) local straight line geodesics (thick segments). The Universe (space) is flat and apparently expanding in an accelerated fashion i.e. redshift z is a linear function of distance d: z K d = .
If that had happened, there would never have been a Big Bang, and there never would have been any "space expansion": no "linear space expansion", no "non-linear space expansion", no "metric space expansion", no "accelerated space expansion, and no "dark energy". The geometry of space on large scales, recovered from the geometry of geodesics, is strongly Lobachevskian, and on local scales can be approximated with arbitrary precision by the Euclidean geometry. "Global scales" and "local scales", as we explained in the text, are determined by the power of our instruments. Nevertheless we see also the presence of matter in the Universe in various states of aggregation, and in constant chaotic motion. From that point of view the Universe is a dynamical system which can be studied by relevant exact mathematical methods.